Annales Mathématiques
Blaise Pascal
no. 1
Hyper–(Abelian–by–finite) groups with many subgroups of finite depth
[Groupes hyper-(Abelien-par-fini) ayant beaucoup de sous-groupes de profondeur finie]
Fares Gherbi1 ; Tarek Rouabhi1
1 Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif, 19000 ALGERIA
Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 17-28.

Le principal résultat de cet article est qu’un groupe G hyper-(Abélien-par-fini) de type fini est fini-par-nilpotent si, et seulement si, toute partie infinie de G contient deux éléments distincts x,y tels que γ n (x,x y )=γ n+1 (x,x y ) pour un certain entier positif n=n(x,y) (respectivement, x,x y est une extension d’un groupe vérifiant la condition minimale sur les sous-groupes normaux par un groupe d’Engel).

The main result of this note is that a finitely generated hyper-(Abelian-by-finite) group G is finite-by-nilpotent if and only if every infinite subset contains two distinct elements x, y such that γ n (x,x y ) =γ n+1 (x,x y ) for some positive integer n=n(x,y) (respectively, x,x y is an extension of a group satisfying the minimal condition on normal subgroups by an Engel group).

DOI : 10.5802/ambp.224
Classification : 20F22, 20F99
Keywords: Infinite subsets, finite depth, Engel groups, minimal condition on normal subgroups, finite-by-nilpotent groups, finitely generated hyper-(Abelian-by-finite) groups
Mot clés : Parties infinies, profondeur finie, Les groupes d’Engel, La condition minimale sur les sous-groupes normaux, les groupes fini-par-nilpotents, les groupes hyper-(Abelien-par-fini) de type fini
Fares Gherbi 1 ; Tarek Rouabhi 1

1 Department of Mathematics Faculty of Sciences Ferhat Abbas University Setif, 19000 ALGERIA 
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Fares Gherbi; Tarek Rouabhi. Hyper–(Abelian–by–finite) groups with many subgroups of finite depth. Annales mathématiques Blaise Pascal, Tome 14 (2007) no. 1, pp. 17-28. doi : 10.5802/ambp.224. https://ambp.centre-mersenne.org/articles/10.5802/ambp.224/

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